5.10 Some Applications of Legendre Polynomials 351
- Solve the potential equation in a hemisphere, 0<ρ<1, 0<φ<π/2,
subject to boundedness conditions atρ=0andφ=0, and the boundary
conditions
u( 1 ,φ)= 1 , 0 <φ<π/ 2 ,
u(ρ, π/ 2 )= 0 , 0 <ρ< 1.
Hint: Use odd-order Legendre polynomials. - Solve this heat problem with convection on a spherical shell of radiusR:
1
R^2 sin(φ)∂
∂φ(
sin(φ)∂∂φu)
−γ^2 (u−T)=^1 k∂∂ut,0 <φ<π, 0 <t,
u(φ, 0 )= 0 , 0 <φ<π.Think carefully about the physical situation before attempting a solution.- Solve this heat problem on a hemispherical shell of radiusR:
1
R^2 sin(φ)∂
∂φ(
sin(φ)∂u
∂φ)
=^1
k∂u
∂t, 0 <φ<π/ 2 , 0 <t,∂u
∂φ(π/ 2 ,t)= 0 , 0 <t,u(φ, 0 )=cos(φ), 0 <φ<π/ 2.- Solve the eigenvalue problem
1
ρ^2[
∂
∂ρ(
ρ^2 ∂u
∂ρ)
+^1
sin(φ)∂
∂φ(
sin(φ)∂u
∂φ)]
=−λ^2 u,0 <ρ<a, 0 <φ<π/ 2 ,
u(a,φ)= 0 , 0 <φ<π/ 2 ,
u(r,π/ 2 )= 0 , 0 <r<a,subject to boundedness conditions atρ=0andatφ=0.- In Part C of this section we mention nodal surfaces (i.e., surfaces where
the function is 0). Find the nodal surfaces of the function
ρ−^1 /^2 J 3 / 2 (λρ)P 1
(
cos(φ))
ifλis the second positive solution ofJ 3 / 2 (λ)=0.- Describe in words the nodal surfaces for
ρ−^1 /^2 J 5 / 2 (λρ)P 2(
cos(φ)